# Questions tagged [stochastic-processes]

A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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### Brownian motion and Durret book

I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...

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### Dependent random variables converging to a density in mean

Let $X$ be an absolutely continuous r.v. with density $f$ which is continuous on $(0,\infty)$. Consider some a.s. decreasing sequence $Y_n$ bounded by $X$ such that $Y_n\searrow 0$ a.s. I stress that ...

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### Existence of stationary stochastic processes with very high correlation

A question was recently asked by a new user, SomeoneHAHA, and then deleted by the user, after receiving an answer. I think the question and the answer (QA) to it may be of interest to some users. ...

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### Angle between Fleming-Viot type 3-particle system

Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...

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### Martingales and intersection of random walks

Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...

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### Extension of probability space problem: Hilbert space valued process V.S. random field

Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field"
Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$
Consider the ...

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### Approximate method to extract behavior of a Laplace transform in an intermediate region

In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...

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### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

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### Upper confidence bound for Poisson process rate parameter

Admittedly, this is an elementary question for mathoverflow. However, I've had no real bites on math and stats.stackexchange so I'm cross-posting.
I am interested in computing an upper confidence ...

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### Stochastic Calculus vs Stochastic Processes in Finance [closed]

I'm a second year student, interested in financial mathematics, who's trying to plan out his degree path currently. There's a stochastic processes unit offered in year 3 and a stochastic calculus unit ...

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### Left passage probability of $SLE_8$?

Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It ...

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### Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...

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### Lebesgue Integral in SDE

In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form
$$
\int_0^t f(s,X(s))ds.
$$
If we ...

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### Bernstein Inequality for continous local martingale

I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.
Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :
$$P(\sup_{t\in [0,T]}...

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### Maximal inequalities of stochastic integral with respect to Poisson random measures

Let $\Phi$ be a smooth convex function such that $\Psi(x)/|x|\to\infty$ as $|x|\to\infty$, and $\hat N(d z,ds)$ be a compensated Poisson measure on $R^d\times R_+$. Do we have the following inequality:...